In geometry, two lines l_1 and l_2 are antiparallel with respect to a given line m if they each make congruent angles with m in opposite senses. More generally, lines l_1 and l_2 are antiparallel with respect to another pair of lines m_1 and m_2 if they are antiparallel with respect to the angle bisector of m_1 and m_2. In any cyclic quadrilateral, any two opposite sides are antiparallel with respect to the other two sides.

Relations

Conic sections

In an oblique cone, there are exactly two families of parallel planes whose sections with the cone are circles. One of these families is parallel to the fixed generating circle and the other is called by Apollonius the subcontrary sections. If one looks at the triangles formed by the diameters of the circular sections (both families) and the vertex of the cone (triangles ABC and ADB), they are all similar. That is, if CB and BD are antiparallel with respect to lines AB and AC, then all sections of the cone parallel to either one of these circles will be circles. This is Book 1, Proposition 5 in Apollonius.